I protest! The suggestion was 4500 Denarii, which was barely supported by any reasoning, 4000 is below any amount mentioned and unsupported by any realistic prediction of the expenses of improving a Province's infrastructure.
What do you expect he gets this money for? Buying groceries? Sending letters home? No, he gets that money to build up the province and such building projects cost a lot of money. 4500 is the absolute minimum I can tolerate for such a venture, and - for the record - I find even that amount insufficient for the task at hand.